The invention is based on a fuel injection pump for internal combustion engines A fuel injection pump of this kind is already known from U.S. Pat. No. 4,737,086. In slide-controlled fuel injection pumps of this kind, the injection quantity and/or onset are determined by the axial location of the control slide. In an in-line pump arrangement, a plurality of such control slides are adjusted in a known manner via a common adjusting shaft, on which there is one transmission element (lever) per pump element. This shaft is located in a conduit extending transversely to the pump axis, and it is rotatably supported on both its ends toward the housing. In order for the transmission element on the adjusting shaft to be capable of engaging a groove of the control slide, a through opening from the transverse conduit to the receiving bore of the pump element comprising the pump piston and pump cylinder must be provided on each engine cylinder. At the same time, this opening also serves to carry fuel between the various pump elements and the transverse conduit, which thus acts as a collective return conduit. The supply onset is regulated via the closure of the filling and relief conduit of the pump piston, by means of the position of the lower edge of the control slide. The end of high-pressure injection (which in terms of time is on the one hand the end of injection and on the other an injection-quantity-determining end of the high-pressure pumping) is defined when the mouth of the filling and relief conduit on the pump piston is uncovered by the control slide, thereby diverting the fuel, which is at very high pressure, from the pump work chamber into the recess of the cylinder liner, via the relief conduit. The diverted fuel stream has very high kinetic energy, which puts a heavy strain on the materials struck by the stream. In the known fuel injection pump of this type, a radial diversion bore is disposed in the control slide; it is opened by an oblique control edge, disposed on the pump piston in the form of a control opening, after the effective injection stroke has been executed. When it is opened, the diversion stream passing through the diversion bore first strikes the wall of the recess of the cylinder liner, where it is reflected, and from there it flows out with high kinetic energy into the aforementioned collective return conduit of the control slide adjusting shaft. This kind of outflow has the disadvantage that the deflected stream, which still has very high kinetic energy, strikes the wall of the suction chamber and flows at high speed past the edges into the collective return conduit. In this process the stream acts upon the fuel housing, which comprises softer material than that of the pump cylinder liner and control slide, which are of tempered steel. The consequence is cavitation and erosion damage on the suction chamber wall of the pump housing and on the edges of the pump housing around which the fuel flows. This damage occurs above all at the transition between the recess and the transverse conduit, because for structural reasons there is only a relatively small transitional cross section, formed from the intersection of the two bores, between the two, with accordingly sharp edges.